Question

(1+ cot a - tan a)(sin a - cos a)/(sec^3 a - cosec^3 a)=sin^2 a .cos^2 a

Answer 1

Step-by-step explanation:

Given -:

>(1 + cotx + tan x)(sinx - cos x) / sec^3 x - cosec^3 x

> (1 + cosx/sinx + sinx/cosx)(sinx - cosx) / 1/cos^3 x - 1/ sin^3 x

>(sin x cos x + sin^2 x + cos^2 x / sinx cos x)(sinx - cos x) / sin^3 x - cos^3x / sin^3xcos^3x

>(sinxcosx + sin^2 x + cos^2 x)(sin x - cos  x)/sinxcosx x sin^3 xcos^3 x / sin^3 x - cos^3 x

  This is in the form of  

>a^3 - b^3 = (a - b)(a^2 + ab + b^2)

So we get

>sin^3 x - cos^3 x / sinx cosx  x  sin^3 x cos^3 x / sin^3 x - cos^3 x

> sin^3 x cos^3 x/sin x cos x

[ = sin^2 x cos^2 x (proved)]

  L.H.S = R.H.S